Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski,^[1] although the idea had been circulating for some decades among mathematicians such as Wacław Sierpiński and António Monteiro.^[2]

A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.

Let ${\displaystyle X}$ be an arbitrary set and ${\displaystyle \wp (X)}$ its power set. A Kuratowski closure operator is a unary operation ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ with the following properties:

[K1] It preserves the empty set: ${\displaystyle \mathbf {c} (\varnothing )=\varnothing }$;

[K2] It is extensive: for all ${\displaystyle A\subseteq X}$, ${\displaystyle A\subseteq \mathbf {c} (A)}$;

[K3] It is idempotent: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {c} (A)=\mathbf {c} (\mathbf {c} (A))}$;

[K4] It preserves binary unions: for all ${\displaystyle A,B\subseteq X}$, ${\displaystyle \mathbf {c} (A\cup B)=\mathbf {c} (A)\cup \mathbf {c} (B)}$.

Because of extensivity [K2], it is possible to weaken the equality in [K3] to a simple inclusion. Furthermore, a consequence of ${\displaystyle \mathbf {c} }$ preserving binary unions is the stronger condition:^[3]

[K4'] It is isotonic: ${\displaystyle A\subseteq B\Rightarrow \mathbf {c} (A)\subseteq \mathbf {c} (B)}$.

It is straightforward to show that weakening the equality in [K4] to a simple inclusion (subadditivity) and assuming [K4'] yields back to [K4] (see the next-to-last paragraph of Proof 2 below).

Kuratowski (1966) includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all ${\displaystyle x\in X}$, ${\displaystyle \mathbf {c} (\{x\})=\{x\}}$. He refers to topological spaces which satisfy all five axioms as T₁-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T₁-spaces via the usual correspondence (see below).^[1]

If requirement [K3] is omitted, then the axioms define a Čech closure operator.^[4] If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator.^[5] A pair ${\displaystyle (X,\mathbf {c} )}$ is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by ${\displaystyle \mathbf {c} }$.

The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:^[6]

[P] For all ${\displaystyle A,B\subseteq X}$, ${\displaystyle A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)\setminus \mathbf {c} (\varnothing )}$.

Axioms [K1]—[K4] can be derived as a consequence of this requirement:

1. Choose ${\displaystyle A=B=\varnothing }$. Then ${\displaystyle \varnothing \cup \mathbf {c} (\varnothing )\cup \mathbf {c} (\mathbf {c} (\varnothing ))=\mathbf {c} (\varnothing )\setminus \mathbf {c} (\varnothing )=\varnothing }$, or ${\displaystyle \mathbf {c} (\varnothing )\cup \mathbf {c} (\mathbf {c} (\varnothing ))=\varnothing }$. This immediately implies [K1].
2. Choose an arbitrary ${\displaystyle A\subseteq X}$ and ${\displaystyle B=\varnothing }$. Then, applying axiom [K1], ${\displaystyle A\cup \mathbf {c} (A)=\mathbf {c} (A)}$, implying [K2].
3. Choose ${\displaystyle A=\varnothing }$ and an arbitrary ${\displaystyle B\subseteq X}$. Then, applying axiom [K1], ${\displaystyle \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (B)}$, which is [K3].
4. Choose arbitrary ${\displaystyle A,B\subseteq X}$. Applying axioms [K1]—[K3], one derives [K4].

Alternatively, A. Monteiro had proposed a weaker axiom that only entails [K2]—[K4]:^[7]

[M] For all ${\displaystyle A,B\subseteq X}$, ${\textstyle A\cup \mathbf {c} (A)\cup \mathbf {c} (\mathbf {c} (B))\subseteq \mathbf {c} (A\cup B)}$.

Requirement [K1] is independent of [M] : indeed, if ${\displaystyle X\neq \varnothing }$, the operator ${\displaystyle \mathbf {c} ^{\star }:\wp (X)\to \wp (X)}$ defined by the constant assignment ${\displaystyle A\mapsto \mathbf {c} ^{\star }(A):=X}$ satisfies [M] but does not preserve the empty set, since ${\displaystyle \mathbf {c} ^{\star }(\varnothing )=X}$. Notice that, by definition, any operator satisfying [M] is a Moore closure operator.

A more symmetric alternative to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]—[K4]:^[2]

[BT] For all ${\displaystyle A,B\subseteq X}$, ${\textstyle A\cup B\cup \mathbf {c} (\mathbf {c} (A))\cup \mathbf {c} (\mathbf {c} (B))=\mathbf {c} (A\cup B)}$.

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map ${\displaystyle \mathbf {i} :\wp (X)\to \wp (X)}$ satisfying the following similar requirements:^[8]

[I1] It preserves the total space: ${\displaystyle \mathbf {i} (X)=X}$;

[I2] It is intensive: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {i} (A)\subseteq A}$;

[I3] It is idempotent: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {i} (\mathbf {i} (A))=\mathbf {i} (A)}$;

[I4] It preserves binary intersections: for all ${\displaystyle A,B\subseteq X}$, ${\displaystyle \mathbf {i} (A\cap B)=\mathbf {i} (A)\cap \mathbf {i} (B)}$.

For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are isotonic, i.e. they satisfy [K4'], and because of intensivity [I2], it is possible to weaken the equality in [I3] to a simple inclusion.

The duality between Kuratowski closures and interiors is provided by the natural complement operator on ${\displaystyle \wp (X)}$, the map ${\displaystyle \mathbf {n} :\wp (X)\to \wp (X)}$ sending ${\displaystyle A\mapsto \mathbf {n} (A):=X\setminus A}$. This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if ${\displaystyle {\mathcal {I}}}$ is an arbitrary set of indices and ${\displaystyle \{A_{i}\}_{i\in {\mathcal {I}}}\subseteq \wp (X)}$,$${\displaystyle \mathbf {n} {\Big (}\bigcup _{i\in {\mathcal {I}}}A_{i}{\Big )}=\bigcap _{i\in {\mathcal {I}}}\mathbf {n} (A_{i}),\qquad \mathbf {n} {\Big (}\bigcap _{i\in {\mathcal {I}}}A_{i}{\Big )}=\bigcup _{i\in {\mathcal {I}}}\mathbf {n} (A_{i}).}$$By employing these laws, together with the defining properties of ${\displaystyle \mathbf {n} }$, one can show that any Kuratowski interior induces a Kuratowski closure (and vice-versa), via the defining relation ${\displaystyle \mathbf {c} :=\mathbf {nin} }$ (and ${\displaystyle \mathbf {i} :=\mathbf {ncn} }$). Every result obtained concerning ${\displaystyle \mathbf {c} }$ may be converted into a result concerning ${\displaystyle \mathbf {i} }$ by employing these relations in conjunction with the properties of the orthocomplementation ${\displaystyle \mathbf {n} }$.

Pervin (1964) further provides analogous axioms for Kuratowski exterior operators^[8] and Kuratowski boundary operators^[9], which also induce Kuratowski closures via the relations ${\displaystyle \mathbf {c} :=\mathbf {ne} }$ and ${\displaystyle \mathbf {c} (A):=A\cup \mathbf {b} (A)}$.

Notice that axioms [K1]—[K4] may be adapted to define an abstract unary operation ${\displaystyle \mathbf {c} :L\to L}$ on a general bounded lattice ${\displaystyle (L,\land ,\lor ,\mathbf {0} ,\mathbf {1} )}$, by formally substituting set-theoretic inclusion with the partial order associated to the lattice, set-theoretic union with the join operation, and set-theoretic intersections with the meet operation; similarly for axioms [I1]—[I4]. If the lattice is orthocomplemented, these two abstract operations induce one another in the usual way. Abstract closure or interior operators can be used to define a generalized topology on the lattice.

Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator ${\displaystyle \mathbf {c} :S\to S}$ on an arbitrary poset ${\displaystyle S}$.

A closure operator naturally induces a topology as follows. Let ${\displaystyle X}$ be an arbitrary set. We shall say that a subset ${\displaystyle C\subseteq X}$ is closed with respect to a Kuratowski closure operator ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ if and only if it is a fixed point of said operator, or in other words it is stable under ${\displaystyle \mathbf {c} }$, i.e. ${\displaystyle \mathbf {c} (C)=C}$. The claim is that the family of all subsets of the total space that are complements of closed sets satisfies the three usual requirements for a topology, or equivalently, the family ${\displaystyle {\mathfrak {S}}[\mathbf {c} ]}$ of all closed sets satisfies the following:

[T1] It is a bounded sublattice of ${\displaystyle \wp (X)}$, i.e. ${\displaystyle X,\varnothing \in {\mathfrak {S}}[\mathbf {c} ]}$;

[T2] It is complete under arbitrary intersections, i.e. if ${\displaystyle {\mathcal {I}}}$ is an arbitrary set of indices and ${\displaystyle \{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]}$, then ${\displaystyle \bigcap _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$;

[T3] It is complete under finite unions, i.e. if ${\displaystyle {\mathcal {I}}}$ is a finite set of indices and ${\displaystyle \{C_{i}\}_{i\in {\mathcal {I}}}\subseteq {\mathfrak {S}}[\mathbf {c} ]}$, then ${\displaystyle \bigcup _{i\in {\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]}$.

Notice that, by extensivity [K2], one may succintly write ${\displaystyle {\mathfrak {S}}[\mathbf {c} ]=\ker(\mathbf {c} -\mathbf {1} )}$, where ${\displaystyle \mathbf {c} -\mathbf {1} :A\mapsto \mathbf {c} (A)\setminus A}$.

Conversely, given a family ${\displaystyle \kappa }$ satisfying axioms [T1]—[T3], it is possible to construct a Kuratowski closure operator in the following way: if ${\displaystyle A\in \wp (X)}$ and ${\displaystyle A^{\uparrow }=\{B\in \wp (X)\ |\ A\subseteq B\}}$ is the inclusion upset of ${\displaystyle A}$, then$${\displaystyle \mathbf {c} _{\kappa }(A):=\bigcap _{B\in (\kappa \cap A^{\uparrow })}B}$$defines a Kuratowski closure operator ${\displaystyle \mathbf {c} _{\kappa }}$ on ${\displaystyle \wp (X)}$.

In fact, these two complementary constructions are inverse to one another: if ${\displaystyle \mathrm {Cls} _{K}(X)}$ is the collection of all Kuratowski closure operators on ${\displaystyle X}$, and ${\displaystyle \mathrm {Atp} (X)}$ is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying [T1]—[T3], then ${\displaystyle {\mathfrak {S}}:\mathrm {Cls} _{K}(X)\to \mathrm {Atp} (X)}$ such that ${\displaystyle \mathbf {c} \mapsto {\mathfrak {S}}[\mathbf {c} ]}$ is a bijection, whose inverse is given by the assignment ${\displaystyle {\mathfrak {C}}:\kappa \mapsto \mathbf {c} _{\kappa }}$.

We observe that one may also extend the bijection ${\displaystyle {\mathfrak {S}}}$ to the collection ${\displaystyle \mathrm {Cls} _{\check {C}}(X)}$ of all Čech closure operators, which strictly contains ${\displaystyle \mathrm {Cls} _{K}(X)}$; this extension ${\displaystyle {\overline {\mathfrak {S}}}}$ is also surjective, which signifies that all Čech closure operators on ${\displaystyle X}$ also induce a topology on ${\displaystyle X}$.^[10] However, this means that ${\displaystyle {\overline {\mathfrak {S}}}}$ is no longer a bijection.

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• If ${\displaystyle X}$ is any set, the operators ${\displaystyle \mathbf {c} _{\bot },\mathbf {c} _{\top }:\wp (X)\to \wp (X)}$ such that $${\displaystyle \mathbf {c} _{\bot }(A)={\begin{cases}\varnothing &A=\varnothing ,\\X&A\neq \varnothing ,\end{cases}}\qquad \mathbf {c} _{\top }(A)=A\quad \forall A\in \wp (X),}$$are Kuratowski closures. The first induces the indiscrete topology ${\displaystyle \{\varnothing ,X\}}$, while the second induces the discrete topology ${\displaystyle \wp (X)}$.
• If ${\displaystyle \lambda }$ is a cardinal number such that ${\displaystyle \lambda \leq \operatorname {crd} (X)}$, then the operator ${\displaystyle \mathbf {c} _{\lambda }:\wp (X)\to \wp (X)}$ such that$${\displaystyle \mathbf {c} _{\lambda }(A)={\begin{cases}A&\operatorname {crd} (A)<\lambda ,\\X&\operatorname {crd} (A)\geq \lambda \end{cases}}}$$satisfies all four Kuratowski axioms.^[11] In the case where e.g. ${\displaystyle \operatorname {crd} (X)={\mathfrak {c}}}$, if ${\displaystyle \lambda =\aleph _{0}}$, this operator induces the cofinite topology on ${\displaystyle X}$; if ${\displaystyle \lambda =\aleph _{1}}$, it induces the cocountable topology.

A pair of Kuratowski closures ${\displaystyle \mathbf {c} _{1},\mathbf {c} _{2}:\wp (X)\to \wp (X)}$ such that ${\displaystyle \mathbf {c} _{2}(A)\subseteq \mathbf {c} _{1}(A)}$ for all ${\displaystyle A\in \wp (X)}$ induce topologies ${\displaystyle \tau _{1},\tau _{2}}$ such that ${\displaystyle \tau _{1}\subseteq \tau _{2}}$, and vice-versa. In other words, ${\displaystyle \mathbf {c} _{1}}$ dominates ${\displaystyle \mathbf {c} _{2}}$ if and only if the topology induced by the latter is a refinement of the topology induced by the former, or equivalently ${\displaystyle {\mathfrak {S}}[\mathbf {c} _{1}]\subseteq {\mathfrak {S}}[\mathbf {c} _{2}]}$.^[12] For example, ${\displaystyle \mathbf {c} _{\bot }}$ clearly dominates ${\displaystyle \mathbf {c} _{\top }}$(the latter just being the identity on ${\displaystyle \wp (X)}$).

In any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to A: ${\displaystyle \mathbf {c} _{A}(B)=A\cap \mathbf {c} _{X}(B)}$, for all ${\displaystyle B\subseteq A}$.^[13]

A function ${\displaystyle f:(X,\mathbf {c} )\to (Y,\mathbf {c} ')}$ is continuous at a point ${\displaystyle p}$ iff ${\displaystyle p\in \mathbf {c} (A)\Rightarrow f(p)\in \mathbf {c} '(f(A))}$, and it is continuous everywhere iff $${\displaystyle f(\mathbf {c} (A))\subseteq \mathbf {c} '(f(A))}$$ for all subsets ${\displaystyle A\in \wp (X)}$.^[14] The mapping ${\displaystyle f}$ is a closed map iff the reverse inclusion holds,^[15] and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.^[16]

A point ${\displaystyle p}$ is close to a subset ${\displaystyle A}$ if ${\displaystyle p\in \mathbf {c} (A).}$This can be used to define a proximity relation on the points and subsets of a set.^[17]

• Topological space
• Closure operator
• Čech closure operator
• Closure algebra
• Characterizations of the category of topological spaces

• Kuratowski, Kazimierz (1966) [1958], Topology, vol. I, translated by Jaworowski, J., Academic Press, ISBN 0-12-429201-1, LCCN 66029221.
• Pervin, William J. (1964), Boas, Ralph P. Jr. (ed.), Foundations of General Topology, Academic Press, ISBN 9781483225159, LCCN 64-17796.
• Arkhangel'skij, A.V.; Fedorchuk, V.V. (1990) [1988], Gamkrelidze, R.V.; Arkhangel'skij, A.V.; Pontryagin, L.S. (eds.), General Topology I, Encyclopaedia of Mathematical Sciences, vol. 17, translated by O'Shea, D.B., Berlin: Springer-Verlag, ISBN 978-3-642-64767-3, LCCN 89-26209.
• Monteiro, António (September 1943), "Caractérisation de l'opération de fermeture par un seul axiome" [Characterization of the operation of closure by a single axiom], Portugaliae mathematica (in French), vol. 4, no. 4 (published 1945), pp. 158–160, Zbl 0060.39406.

• Alternative Characterizations of Topological Spaces